Number Bases (Radices)

In mathematical numeral systems, the radix or base is the number of unique digits, including zero, used to represent numbers in a positional numeral system. For example, for the decimal system (the most common system in use today) the radix is ten, because it uses the ten digits from 0 through 9. Source

Today I turned 24793 days old, a prime number of days, and as it turns out a palindrome in base 12 (12421). I started playing around with the representation of 24793 in other number bases or radices. In base 31, the representation is poo so I may be in for a shitty day. In base 36, the letters of the alphabet are exhausted:

Beyond base 36, the following system is used:

More generally, in a system with radix b (b > 1), a string of digits $\ d_1 … d_n$ denotes the number $\ d_1b^{n−1}+d_2b^{n−2}+ … +d_nb^{0}$, where $\ 0 ≤ d_i < b$.

In practice, a colon is used to separate the individual digits e.g. 24793 is written$\ 18:4:3 \,_{37}$  in base 37 which can be dispensed with of course in the case of base 10.

Radices are usually natural numbers but they don't have to be. For example, a base using the golden ratio$\ \Phi$ is possible, remembering that$\ \Phi$ is given by the expression: $$\ \frac{1+\sqrt{5}}{2}$$ Commonly, the capital letter$\ \Phi$ is used to represent 1.618033988749895… and the lower case letter $\phi$ to represent 0.618033988749895… or$\ \Phi-1$ but this is certainly not always the case.

Such a base is colloquially called phinary and the following table gives some idea of how it works (source) but uses$\ \varphi$, a variation of the lowercase$\ \phi$ but meant to equal 1.618033988749895… here:

$\ \Phi$ is closely linked to the Fibonacci sequence since $$\lim_{n \to \infty} \frac{F_n}{F_{n-1}}=\Phi$$ More information about $\Phi$ as a number base can be found on this site.